Riemann zeta function: Zeros (formula 10.01.30.0008)

Zeta

$Mathematica Notation$

Zeta Functions and Polylogarithms

Zeta[s]

Zeros

Sums over zeros

http://functions.wolfram.com/10.01.30.0008.01

Input Form

Subscript[Î–,8]ï±2*StieltjesGamma[1]^4 + (4/3)*StieltjesGamma[3]* StieltjesGamma[1]^2 + 8*EulerGamma^6*StieltjesGamma[1] + 2*StieltjesGamma[2]^2*StieltjesGamma[1] + (StieltjesGamma[5]*StieltjesGamma[1])/15 + EulerGamma^8 + (16*StieltjesGamma[1]^3 + 4*StieltjesGamma[3]* StieltjesGamma[1] + 3*StieltjesGamma[2]^2 + StieltjesGamma[5]/15)*EulerGamma^2 + StieltjesGamma[3]^2/9 + 4*EulerGamma^5* StieltjesGamma[2] + (4/3)*EulerGamma^4* (15*StieltjesGamma[1]^2 + StieltjesGamma[3]) + (StieltjesGamma[2]*StieltjesGamma[4])/6 + (1/3)*EulerGamma^3*(48*StieltjesGamma[1]* StieltjesGamma[2] + StieltjesGamma[4]) + (1/90)*EulerGamma* (60*(2*StieltjesGamma[2]*(9*StieltjesGamma[1]^2 + StieltjesGamma[3]) + StieltjesGamma[1]* StieltjesGamma[4]) + StieltjesGamma[6]) + StieltjesGamma[7]/630 - (17*Pi^8)/161280 + 1/; Subscript[Î–,8]ï±Limit[Sum[1/Subscript[Ï,k]^8,Abs[Subscript[Ï,k]<T]], Tï”¢Infinity]&&Zeta[Subscript[Ï,k]]ï±0&& Re[Subscript[Ï,k]]â‰ 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[CapitalZeta]", "8"], "\[Equal]", RowBox[List["1", "+", SuperscriptBox["EulerGamma", "8"], "-", FractionBox[RowBox[List["17", " ", SuperscriptBox["\[Pi]", "8"]]], "161280"], "+", RowBox[List["8", " ", SuperscriptBox["EulerGamma", "6"], " ", RowBox[List["StieltjesGamma", "[", "1", "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["StieltjesGamma", "[", "1", "]"]], "4"]]], "+", RowBox[List["4", " ", SuperscriptBox["EulerGamma", "5"], " ", RowBox[List["StieltjesGamma", "[", "2", "]"]]]], "+", RowBox[List["2", " ", RowBox[List["StieltjesGamma", "[", "1", "]"]], " ", SuperscriptBox[RowBox[List["StieltjesGamma", "[", "2", "]"]], "2"]]], "+", RowBox[List[FractionBox["4", "3"], " ", SuperscriptBox[RowBox[List["StieltjesGamma", "[", "1", "]"]], "2"], " ", RowBox[List["StieltjesGamma", "[", "3", "]"]]]], "+", FractionBox[SuperscriptBox[RowBox[List["StieltjesGamma", "[", "3", "]"]], "2"], "9"], "+", RowBox[List[FractionBox["4", "3"], " ", SuperscriptBox["EulerGamma", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["15", " ", SuperscriptBox[RowBox[List["StieltjesGamma", "[", "1", "]"]], "2"]]], "+", RowBox[List["StieltjesGamma", "[", "3", "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["StieltjesGamma", "[", "2", "]"]], " ", RowBox[List["StieltjesGamma", "[", "4", "]"]]]], "+", RowBox[List[FractionBox["1", "3"], " ", SuperscriptBox["EulerGamma", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["48", " ", RowBox[List["StieltjesGamma", "[", "1", "]"]], " ", RowBox[List["StieltjesGamma", "[", "2", "]"]]]], "+", RowBox[List["StieltjesGamma", "[", "4", "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["EulerGamma", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["16", " ", SuperscriptBox[RowBox[List["StieltjesGamma", "[", "1", "]"]], "3"]]], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["StieltjesGamma", "[", "2", "]"]], "2"]]], "+", RowBox[List["4", " ", RowBox[List["StieltjesGamma", "[", "1", "]"]], " ", RowBox[List["StieltjesGamma", "[", "3", "]"]]]], "+", FractionBox[RowBox[List["StieltjesGamma", "[", "5", "]"]], "15"]]], ")"]]]], "+", RowBox[List[FractionBox["1", "15"], " ", RowBox[List["StieltjesGamma", "[", "1", "]"]], " ", RowBox[List["StieltjesGamma", "[", "5", "]"]]]], "+", RowBox[List[FractionBox["1", "90"], " ", "EulerGamma", " ", RowBox[List["(", RowBox[List[RowBox[List["60", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["StieltjesGamma", "[", "2", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["9", " ", SuperscriptBox[RowBox[List["StieltjesGamma", "[", "1", "]"]], "2"]]], "+", RowBox[List["StieltjesGamma", "[", "3", "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["StieltjesGamma", "[", "1", "]"]], " ", RowBox[List["StieltjesGamma", "[", "4", "]"]]]]]], ")"]]]], "+", RowBox[List["StieltjesGamma", "[", "6", "]"]]]], ")"]]]], "+", FractionBox[RowBox[List["StieltjesGamma", "[", "7", "]"]], "630"]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["\[CapitalZeta]", "8"], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List[UnderscriptBox["\[Sum]", RowBox[List[RowBox[List["\[LeftBracketingBar]", SubscriptBox["\[Rho]", "k"], "\[RightBracketingBar]"]], "<", "T"]]], FractionBox["1", SubsuperscriptBox["\[Rho]", "k", "8"]]]], ",", RowBox[List["T", "\[Rule]", "\[Infinity]"]]]], "]"]]]], "\[And]", RowBox[List[RowBox[List["Zeta", "[", SubscriptBox["\[Rho]", "k"], "]"]], "\[Equal]", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", SubscriptBox["\[Rho]", "k"], "]"]], "\[NotEqual]", "0"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <mi> Ζ </mi> <mn> 8 </mn> </msub> <mo> ⩵ </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msubsup> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 1 </mn> <mn> 4 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 4 </mn> <mn> 3 </mn> </mfrac> <mo> ⁢ </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 3 </mn> </msub> <mo> ⁢ </mo> <msubsup> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <msup> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", 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</mn> </mfrac> <mo> ⁢ </mo> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[EulerGamma]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 60 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 9 </mn> <mo> ⁢ </mo> <msubsup> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> 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<semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mo> ( </mo> <mrow> <munder> <mo> ∑ </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> ρ </mi> <mi> k </mi> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> = </mo> <mn> 1 </mn> </mrow> </munder> <mfrac> <mn> 1 </mn> <msubsup> <mi> ρ </mi> <mi> k </mi> <mn> 8 </mn> </msubsup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> ρ </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[SubscriptBox["\[Rho]", "k"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, 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StieltjesGamma </ci> </apply> <ms> 1 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 3 </ms> <apply> <ci> SubsuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 2 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 5 </ms> </apply> <ms> 15 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> ℽ </ms> <apply> <ci> Function </ci> <eulergamma /> </apply> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <apply> <ci> SubsuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 3 </ms> <ms> 2 </ms> </apply> <ms> 9 </ms> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <apply> <ci> SuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> ℽ </ms> <apply> <ci> Function </ci> <eulergamma /> </apply> </apply> <ms> 5 </ms> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 4 </ms> <ms> 3 </ms> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> ℽ </ms> <apply> <ci> Function </ci> <eulergamma /> </apply> </apply> <ms> 4 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 15 </ms> <apply> <ci> SubsuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 3 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 2 </ms> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 4 </ms> </apply> </list> </apply> <ms> 6 </ms> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 3 </ms> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> ℽ </ms> <apply> <ci> Function </ci> <eulergamma /> </apply> </apply> <ms> 3 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 48 </ms> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 1 </ms> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 4 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 90 </ms> </apply> <apply> <ci> TagBox </ci> <ms> ℽ </ms> <apply> <ci> Function </ci> <eulergamma /> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 60 </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 2 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 9 </ms> <apply> <ci> SubsuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 3 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 1 </ms> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 4 </ms> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 6 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 7 </ms> </apply> <ms> 630 </ms> </apply> <ms> - </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 17 </ms> <apply> <ci> SuperscriptBox </ci> <ms> π </ms> <ms> 8 </ms> </apply> </list> </apply> <ms> 161280 </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> Ζ </ms> <ms> 8 </ms> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <ms> Limit </ms> <ms> [ </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> ∑ </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms>  </ms> <apply> <ci> SubscriptBox </ci> <ms> ρ </ms> <ms> k </ms> </apply> <ms>  </ms> </list> </apply> <ms> = </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> SubsuperscriptBox </ci> <ms> ρ </ms> <ms> k </ms> <ms> 8 </ms> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> T </ms> <ms> -> </ms> <ms> ∞ </ms> </list> </apply> <ms> ] </ms> </list> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ζ </ms> <ms> ( </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> ρ </ms> <ms> k </ms> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <ci> BoxForm`e$ </ci> <apply> <ci> Zeta </ci> <ci> BoxForm`e$ </ci> </apply> </apply> </apply> </apply> <ms> ⩵ </ms> <ms> 0 </ms> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> ρ </ms> <ms> k </ms> </apply> <ms> ) </ms> </list> </apply> <ms> ≠ </ms> <ms> 0 </ms> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>

Date Added to functions.wolfram.com (modification date)

2003-08-21

Zeta[s,a]